Lightning Calculation by 9-Digit Number

Party Trick

Ask someone in the audience for some $9$-digit number.

You will then announce that you are going to multiply it by $142 \, 857 \, 143$ in your head.

You start writing down that product from the moment the first few digits are being given to you.

You mentally write down that big number twice, concatenating the two parts to obtain a big number of twice the length.

Then you divide this number by $7$, which, using the standard algorithm for division, you do from the left hand side.

The showman announces he will instantly multiply any given $9$-digit number by $142 \, 857 \, 143$.

The given $9$-digit number is $577 \, 831 \, 345$.

Thus by division by $7$:

   82,547,335,082,547,335
 ------------------------
7)577,831,345,577,831,345

We have that:

$142 \, 857 \, 143 \times 7 = 1 \, 000 \, 000 \, 001$

By writing down the given $9$-digit number twice, you are multiplying it by $1 \, 000 \, 000 \, 001$.

So dividing that number by $7$ you are multiplying the original $9$-digit number by $142 \, 857 \, 143$.